Wave regimes on power-law fluid film flowing down a porous plane

Non-linear waves on the surface of a falling film of power-law fluid on a vertical porous plane are investigated. The waves are described by evolution equations generalising equations previously derived in the case of solid plane. It is shown that the slip condition on the interface between pure liquid and the porous substrate drastically changes structure of the steady waves travelling in the film.     

Bifurcation of limit cycles in fluid film bearings

Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation.     

Waves on the surface of a falling power-law fluid film

Waves that occur at the surface of a falling film of thin power-law fluid on a vertical plane are investigated. Using the method of integral relations an evolution equation is derived for two types of waves equation which are possible under long wave approximation. This equation reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and the energy is acquired from the mean flow during the interaction of the waves, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. It is also found that this exchange of energy between kinematic and inertial waves strongly depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that the surface tension plays a double role: for a kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for a dynamic wave process it yields dispersion. Further, it is shown that the non-Newtonian character n plays a vital role in controlling the role of the term that contains surface tension in the above processes.     

Weakly nonlinear stability analysis of a falling film with countercurrent gas flow

Weakly nonlinear stability analysis of a falling film with countercurrent gas–liquid flow has been investigated. A normal mode approach and the method of multiple scales are employed to carry out the linear and nonlinear stability solutions for the film flow system. The results show that both supercritical stability and subcritical instability are possible for a film flow system when the gas flows in the countercurrent direction. The stability characteristics of the film flow system are strongly influenced by the effects of interfacial shear stress when the gas flows in the countercurrent direction. The effect of countercurrent gas flow in a falling film is to stabilize the film flow system.     

New solutions for surface tension driven spreading of a thin film

The standard fourth-order non-linear PDE modelling the flow of thin fluid film subject to surface tension is studied. The Lie group method is used to reduce the model equation from a fourth-order PDE to a fourth-order ODE. Analytical solutions are obtained for certain cases. Where analytical progress cannot be made, we determine numerical solutions.     

Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt

In this paper, we prove in general that the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the homotopy analysis method (HAM) profound in 1992 when ħ = −1. Besides, by using the thin film flows of Sisko and Oldroyd 6-constant fluids on a moving belt as examples, we show that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.: Chaos Solitons and Fractals (2006) in press) are divergent, and thus useless. However, by choosing a proper value of the auxiliary parameter ħ, we give convergent series solution by means of the HAM. These two examples also show that, different from the HPM and other traditional analytic techniques, the HAM indeed provides us with a simple way to ensure the convergence of the solution.     

Development of a PLIC-VOF method for the dynamic simulation of entry region flow in a laminar falling film

The present work describes a numerical procedure to simulate the development of hydrodynamic entry region in a gravity-driven laminar liquid film flow over an inclined plane. It provides a better insight into the physics of developing film in entry region. A novel numerical approach is proposed which has the potential to provide solutions for the complex physics of liquid film spreading on solid walls. The method employs an incompressible flow algorithm to solve the governing equations, a PLIC-VOF method to capture the free surface evolution and a continuum surface force (CSF) model to include the effect of surface tension. To account for the moving contact line on the solid substrate, a precursor film model based wall treatment is implemented. Liquid film flow has been simulated for the Reynolds number range of 5 ≤ Re ≤ 37.5, and the predicted results are found to agree well with the available analytical and experimental data.     

Structure of permanent waves in density-stratified media

Non-linear steady waves in density-stratified channels are discussed. Under the assumption of non-diffusive incompressible flow, a method to classify all waveforms is given for moderate wave amplitudes, when the densityq at infinity decreases with altitude (q<0). The classification is worked out explicitly for the two highest critical Froude numbers. Furthermore, special density distributions are constructed with the purpose of showing that, for example, solitary waves exist even ifq>0.Dedicated to Professor Gaetano Fichera on the occasion of his 70th birthday. General lecture delivered at the 11th Italian National Congress of Theoretical and Applied Mechanics (AIMETA), Trento, September 1992.     

Wave regimes on a film of generalized Newtonian fluid flowing down a vertical plane

The flow of a thin film of generalized Newtonian fluid down a vertical wall in the gravity field is considered. For small flow-rates, in the long-wave approximation, an equation describing the evolution of the surface perturbations is obtained. Depending on the signs of the coefficients, this equation is equivalent to one of four equations with solutions significantly different in evolutionary behavior. For the most interesting case, soliton solutions are numerically found.     

Simulation and measurement of 3D shear-driven thin liquid film flow in a duct

Three-dimensional flow behavior of thin liquid film that is shear-driven by turbulent air flow in a duct is measured and simulated. Its film thickness and width are reported as a function of air velocity, liquid flow rate, surface tension coefficient, and wall contact angle. The numerical component of this study is aimed at exploring and assessing the suitability of utilizing the FLUENT-CFD code and its existing components, i.e. Volume of Fluid model (VOF) along with selected turbulence model, for simulating the behavior of 3D shear-driven liquid film flow, through a comparison with measured results. The thickness and width of the shear-driven liquid film are measured using an interferometric technique that makes use of the phase shift between the reflections of incident light from the top and bottom surfaces of the thin liquid film. Such measurements are quite challenging due to the dynamic interfacial instabilities that develop in this flow. The results reveal that higher air flow velocity decreases the liquid film thickness but increases its width, while higher liquid flow rate increases both its thickness and width. Simulated results provide good estimates of the measured values, and reveal the need for considering a dynamic rather than a static wall contact angle in the model for improving the comparison with measured values.     

A new solution for the rotation-driven spreading of a thin fluid film

Lie groups are used to solve the equation governing the flow of a thin liquid film subject to centrifugal spreading and viscous resistance. A new implicit solution is found. It is shown how this relates to the previous known solutions for the spreading of an initially flat film, the steady state and a separable solution. New permissible forms for the film evolution are also studied, including solutions exhibiting finite time blow-up. Near the contact line, where the film height tends to zero, an approximate explicit solution is obtained which may be used to describe a film with any size contact angle.     

Influence of the capillarity on a creeping film flow down an inclined plane with an edge

Summary In creeping flows of thin films, the capillarity can play a dominant role. In this paper, the creeping film flow down an inclined plane with an edge is considered. The influence of the capillarity on the velocity and the film surface is studied analytically, numerically and experimentally. Received 12 April 1999; accepted for publication 9 May 1999     

Nonlinear analyses of wrinkles in a film bonded to a compliant substrate

Subject to a compressive membrane force, a film bonded to a compliant substrate often forms a pattern of wrinkles. This paper studies such wrinkles in a layered structure used in several recent experiments. The structure comprises a stiff film bonded to a compliant substrate, which in turn is bonded to a rigid support. Two types of analyses are performed. First, for sinusoidal wrinkles, by minimizing energy, we obtain the wavelength and the amplitude of the wrinkles for substrates of various moduli and thicknesses. Second, we develop a method to simultaneously evolve the two-dimensional pattern in the film and the three-dimensional elastic field in the substrate. The simulations show that the wrinkles can evolve into stripes, labyrinths, or herringbones, depending on the anisotropy of the membrane forces. Statistical averages of the amplitude and wavelength of wrinkles of various patterns correlate well with the analytical solution of the sinusoidal wrinkles.     

Numerical study of the parametric evolution of bifurcations in the three-dimensional ring problem of N bodies

We study the dynamics of a massless particle in an annular configuration of N bodies, N − 1 of which have equal masses m and are located in equal distances on a fictitious circle and one has mass βm and is located at the center of the circle. Our interest is focused on the bifurcation points from planar to three-dimensional families of symmetric periodic orbits in the above problem. We study numerically the evolution of these bifurcation points with respect to the variation of the mass parameter β. In particular we investigate the continuous evolution of bifurcation points for values of β from 2 up to 1000. The two distinct cases of the system’s behavior at β = 2 and 1000 are examined comparatively and various conclusions are drawn regarding the overall dynamical evolution of the three-dimensional system as the relative mass of the central body grows.     

The Mexican hat effect on the delamination buckling of a compressed thin film

Because of the interaction between film and substrate,the film buckling stress can vary significantly,depending on the delamination geometry,the film and substrate mechanical properties.The Mexican hat effect indicates such interaction.An analytical method is presented,and related dimensional analysis shows that a single dimensionless parameter can effectively evaluate the effect.     

Effects of air-flow and evaporation on the development of thin liquid film over a rotating annular disk

Axisymmetric flow of thin pure liquid film on a spinning horizontal annular disk is studied under the action of air shear at the liquid–air interface and evaporation. The non-linear evolution equation that is obtained by singular perturbation method is solved analytically, for small Reynolds number, by using the method of characteristic and numerically by the use of Newton–Kantorovich method for any Reynolds number. Font breakdown time and its location from the center of the disk is predicted both by analytically and numerically. The result shows that the thinning of the initial film increases as air stress increase, same result is also escalated in presence of evaporation.     

Temperature pulsations and thermocapillary effects on the surface of a wavy heated liquid film

The temperature pulsations and wave characteristics in water film flow along a vertical plate with a heater are investigated. Using an infrared scanner, the temperature field on the film surface is measured for various heat flux densities on the heater. Experimental data on the variation of the temperature with time on a local segment of the liquid film surface during wave transmission are obtained. In the absence of a heat flux the data obtained are in good agreement with the results of other researchers for an isothermal liquid film. When the down-flowing liquid is heated, the thermocapillary forces lead to the formation of rivulets and a thin film between them. It is shown that in the inter-rivulet zone the relative wave amplitude increases due to the action of the thermocapillary forces.     

A computational model for the indentation and phase transformation of a martensitic thin film

We propose a computational model for a stress-induced martensitic phase transformation of a single-crystal thin film by indentation and its reverse transformation to austenite by heating. Our model utilizes a surface energy that allows sharp interfaces with finite energy and a penalty that forces the film to lie above the indenter and undergo a stress-induced austenite-to-martensite phase transformation. We introduce a method to nucleate the martensite-to-austenite phase transformation since in our model the film would otherwise remain in the martensitic phase in a local minimum of the energy.     

The numerical solution of the thin film flow surrounding a horizontal cylinder resulting from a vertical cylindrical jet

The numerical solution of the thin film flow surrounding a horizontal cylinder resulting from a single vertical cylindrical jet is obtained. This is effected by transforming the domain of the flow, which contains a free surface, onto a rectangular parallelepiped and using a marching strategy to solve the ensuing parabolic equations. The flow terminates at a finite distance along the cylinder, its position depending on the velocity and mass flux of the jet. A comparison with the usual two-dimensional model in which the jet is replaced by a vertical sheet shows that such a representation is valid provided the overall width of the flow is not too large. In particular, the differences in heat transfer characteristics amount to a few per cent, thus validating the use of the two-dimensional model when applied to heat exchanger tubes. A comparison with the more usual multicolumn case is also considered.